### Nuprl Lemma : decidable__rel_exp

`∀k,n:ℕ.  ∀[R:ℕn ⟶ ℕn ⟶ ℙ]. ((∀i,j:ℕn.  Dec(i R j)) `` (∀i,j:ℕn.  Dec(i R^k j)))`

Proof

Definitions occuring in Statement :  rel_exp: `R^n` int_seg: `{i..j-}` nat: `ℕ` decidable: `Dec(P)` uall: `∀[x:A]. B[x]` prop: `ℙ` infix_ap: `x f y` all: `∀x:A. B[x]` implies: `P `` Q` function: `x:A ⟶ B[x]` natural_number: `\$n`
Definitions unfolded in proof :  all: `∀x:A. B[x]` rel_exp: `R^n` eq_int: `(i =z j)` subtract: `n - m` ifthenelse: `if b then t else f fi ` btrue: `tt` infix_ap: `x f y` uall: `∀[x:A]. B[x]` implies: `P `` Q` member: `t ∈ T` nat: `ℕ` prop: `ℙ` so_lambda: `λ2x.t[x]` so_apply: `x[s]` bool: `𝔹` unit: `Unit` it: `⋅` uiff: `uiff(P;Q)` and: `P ∧ Q` uimplies: `b supposing a` false: `False` guard: `{T}` bfalse: `ff` exists: `∃x:A. B[x]` or: `P ∨ Q` sq_type: `SQType(T)` bnot: `¬bb` assert: `↑b` subtype_rel: `A ⊆r B` nequal: `a ≠ b ∈ T ` decidable: `Dec(P)` iff: `P `⇐⇒` Q` not: `¬A` rev_implies: `P `` Q` le: `A ≤ B` less_than': `less_than'(a;b)` true: `True` top: `Top`
Lemmas referenced :  decidable__equal_int_seg int_seg_wf all_wf decidable_wf eq_int_wf bool_wf eqtt_to_assert assert_of_eq_int less_than_transitivity1 le_weakening less_than_irreflexivity eqff_to_assert equal_wf bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot neg_assert_of_eq_int decidable__exists_int_seg infix_ap_wf rel_exp_wf subtract_wf decidable__le false_wf not-le-2 not-equal-2 less-iff-le condition-implies-le minus-one-mul zero-add minus-one-mul-top minus-add minus-minus add-associates add-swap add-commutes add_functionality_wrt_le add-zero le-add-cancel le_wf decidable__and2 nat_wf uall_wf set_wf less_than_wf primrec-wf2
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation cut thin sqequalRule isect_memberFormation introduction extract_by_obid sqequalHypSubstitution dependent_functionElimination natural_numberEquality setElimination rename hypothesisEquality hypothesis isectElimination lambdaEquality because_Cache applyEquality functionExtensionality functionEquality cumulativity universeEquality unionElimination equalityElimination equalityTransitivity equalitySymmetry productElimination independent_isectElimination independent_functionElimination voidElimination dependent_pairFormation promote_hyp instantiate productEquality dependent_set_memberEquality independent_pairFormation addEquality minusEquality isect_memberEquality voidEquality intEquality

Latex:
\mforall{}k,n:\mBbbN{}.    \mforall{}[R:\mBbbN{}n  {}\mrightarrow{}  \mBbbN{}n  {}\mrightarrow{}  \mBbbP{}].  ((\mforall{}i,j:\mBbbN{}n.    Dec(i  R  j))  {}\mRightarrow{}  (\mforall{}i,j:\mBbbN{}n.    Dec(i  rel\_exp(\mBbbN{}n;  R;  k)  j)))

Date html generated: 2017_04_14-AM-07_38_07
Last ObjectModification: 2017_02_27-PM-03_10_13

Theory : relations

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